On digits of Mersenne numbers

Motivated by recently developed interest to the distribution of $q$-arydigits of Mersenne numbers $M_p = 2^p-1$, where $p$ is prime, we estimaterational exponential sums with $M_p$, $p \leq X$, modulo a large power of afixed odd prime $q$. In turn this immediately implies the normality of stringsof $q$-ary digits amongst about $(\log X)^{3/2+o(1)}$ rightmost digits of$M_p$, $p \leq X$. Previous results imply this only for about (\logX)^{1+o(1)} rightmost digits.<br

Microstructuration of poly(3-hexylthiophene) leads to bifunctional superhydrophobic and photoreactive surfaces

Schematic representation and a preparation route for the poly(3-hexylthiophene) conducting polymer film having both superhydrophobic and visible-light active photocatalytic properties.</p

On lattice profile of the elliptic curve linear congruential generators

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG

On the dynamical system generated by the Möbius transformation at prime times

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map x↦ (ax+ b) / (cx+ d) over a finite field of p elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences

Multiplicative and Linear Dependence in Finite Fields and on Elliptic Curves Modulo Primes

For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure (F) over bar (p) of a finite prime field F-p, as well as L-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions phi(1), ... , phi(m), rho(1), ... , rho(n) is an element of Q(X) and an elliptic curve E defined over the rational numbers Q, for any sufficiently large prime p, for all but finitely many alpha is an element of (F) over bar (p), at most one of the following two can happen: phi(1)(alpha), ... , phi(m)(alpha) are K-multiplicatively dependent or the points (rho(1)(alpha), .), ... , (rho(n)(alpha), .) are L-linearly dependent on the reduction of E modulo p. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety G(m)(m) x E-n with the algebraic subgroups of codimension at least 2.As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases